2007_Further results on Lyapunov functions for slowly time-varying systems.pdf

7/25/2019 2007_Further results on Lyapunov functions for slowly time-varying systems.pdf

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Math. Control Signals Syst. (2007) 19:121DOI 10.1007/s00498-006-0010-4

O R I G I NA L A RT I C L E

Further results on Lyapunov functions for slowly

time-varying systems

Frdric Mazenc Michael Malisoff

Received: 4 April 2006 / Accepted: 20 November 2006 /

Published online: 4 January 2007 Springer-Verlag London Limited 2007

Abstract We provide general methods for explicitly constructing strictLyapunov functions for fully nonlinear slowly time-varying systems. Our resultsapply to cases where the given dynamics and corresponding frozen dynamics arenot necessarily exponentially stable. This complements our previous Lyapunovfunction constructions for rapidly time-varying dynamics. We also explicitly con-struct input-to-state stable Lyapunov functions for slowly time-varying control

systems. We illustrate our findings by constructing explicit Lyapunov functionsfor a pendulum model, an example from identification theory, and a perturbedfriction model.

Keywords Lyapunov function constructionsSlowly time-varying systemsStability analysisInput-to-state stability

1 Introduction

This paper is devoted to the study of fully nonlinear slowly time-varying systemsof the form

x=f(x, t, t/) (1)

F. MazencUMR Analyse des Systmes et Biomtrie INRA, 2,

pl. Viala, 34060 Montpellier, Francee-mail: [email protected]

M. Malisoff (B)Department of Mathematics, Louisiana State University,Baton Rouge, LA 70803-4918, USAe-mail: [email protected]


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2 F. Mazenc, M. Malisoff

for large values of the constant > 0 (but see Sect. 7 for the extension tosystems with controls). See Sect.2for our standing assumptions on (1). Suchsystems arise in a large variety of important engineering applications such asthe control of friction and pendulums [6, 13, 14]. It is therefore of great inter-

est in control engineering to develop methods for determining whether slowlytime-varying systems are uniformly globally asymptotically stable (UGAS).When (1) is UGAS, it is also highly desirable to have general methods forconstructing explicit closed form Lyapunov functions for (1). See for exam-ple[13, 911] for discussions on the essentialness of Lyapunov functions forfeedback design and robustness analysis. See also [11, 12] for the dual problemof stabilizingrapidlytime-varying systems, and see Remark8for the relation-ship between our methods for constructing Lyapunov functions for rapidly andslowly time-varying systems.

One popular approach to studying (1) is to first establish exponential stabilityof the corresponding frozen dynamics

x=f(x, t, ) (2)

for all relevant values of the parameterincluding cases where the exponentin the exponential decay estimate can be negative or positive for some valuesofbut is positive on average[6, 13,14]. The stability of the frozen dynamics

is then used to establish stability of (1). However, these earlier results do notlead to explicit Lyapunov functions for (1) that would be needed for robust-ness analysis. The main objectives of our work are (i) to show that explicitLyapunov functions for (1) can be explicitly constructed in terms of a suitableclass of oftentimes readily available Lyapunov functions for (2) when the con-stant >0 is large enough, and (ii) to show how to relax the exponential likestability assumptions on (2) and also allow cases whereis a vector, therebybroadening the class of dynamics to which the frozen dynamics method can beapplied.

The rest of this paper is organized as follows. In Sect.2 we provide the relevantdefinitions and standing assumptions on (1). We state and prove our main resultin Sects.34.In Sect.5,we extend our main result to cases where the Lyapunovfunctions for the frozen systems satisfy less restrictive properties than those inSects.34.In Sect.6,we illustrate the wide applicability of our results usingfour examples. In the first two examples, the family of Lyapunov like functionsfor (2) is independent of, so the strict Lyapunov functions we construct for(1) are valid for all >0, i.e. (1) is UGAS for all >0. Our next two examplesinvolve a mass spring model with slowly time-varying coefficients from [4] andan identification model similar to those studied in [12], and illustrate the moregeneral situation where (1) is not necessarily UGAS for all values of >0. Ineach case, the dynamics have slowly time-varying coefficients and so are beyondthe scope of the previously known Lyapunov construction methods. In Sect.7,we show how to extend our results to systems with controls using input-to-statestability. We close in Sect.8with some suggestions for further research.


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Lyapunov functions for slowly time-varying systems 3

2 Definitions, assumptions, and lemmas

We let K denote the set of all continuous functions :[0, ) [0, )forwhich (i) (0)

=0 and (ii) is strictly increasing and unbounded. We letKL

denote the class of all continuous functions :[0, ) [0, ) [0, ) forwhich

(I) (, t) Kfor eacht0,(II) (s, )is non-increasing for eachs0, and

(III) (s, t)0 ast +for eachs0.When we say that a function is smooth (a.k.a.C1), we mean that it is con-tinuously differentiable, written C1. (For functions defined on[0, ),we interpret(0) as a one-sided derivative, and continuity of at 0 as one-sided continuity.) We let

| |denote the Euclidean norm. A continuous function

:[0, ) [0, )is calledpositive definite provided it is zero only at zero.Whenrp(r) Rd is a function with differentiable components, we usep(r)to denote the vector(p1(r), . . . ,p

d(r)).

The following definitions and lemma apply to general nonlinear systems

x=h(x, t) (3)

evolving on the state spaceRn where h is locally Lipschitz (but see Sect. 7 for theextension to control systems). Later we specialize to systems with multiple time

scales and frozen parameters, e.g.h(x, t)=f(x, t,p(t/)) or h(x, t)=f(x, t, ) forgiven constant parameters and and suitable functionsp. We always assumethat (3) isforward completemeaning for eachx0 Rn andt0 R0 := [0, )there exists a unique trajectory

[t0, ) t (t; t0,x0)

for (3) that satisfiesx(t0)=x0. We assume that all of our uncontrolled dynamics(3) areuniformly state boundedmeaning that there existsh K such that|h(x, t)| h(|x|)everywhere.Definition 1 Wesaythat(3) is uniformly globally asymptotically stable (UGAS)provided that there exists KL such that

| (t; t0,x0)| (|x0|, t t0) (UGAS)

for allx0 Rn,t0 [0, ), andtt0.Definition 2 A smooth function W: Rn [0, ) [0, ) is called a Lyapunovfunction for (3) provided that there are functions1, 2

K

and a positive

definite function3such that

(L1) 1(|x|)W(x, t)2(|x|) and(L2) Wt(x, t) + Wx(x, t)h(x, t) 3(|x|)hold for allt0 andx Rn.


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The subscripts onWdenote partial gradients. In what follows, we often omitthe argumentsx,t, etc., in our functions when they are clear from the context;and all (in)equalities should be interpreted to hold wherever they make sense.A smooth function W : Rn

[0,

)

[0,

) that admits 1, 2

K

such

that (L1) holds everywhere is calleduniformly proper and positive definite. Thefollowing lemma is standard [5, 6]:

Lemma 3 If(3)admits a Lyapunov function, then it is UGAS.

A simple application of Fubinis Theorem yields the formula

ttc

ts

(l) dlds=t

tc(r t+ c)(r)dr

and therefore also the following [10]:

Lemma 4 Let: R R be continuous and bounded in norm by some constantM>0, and c> 0 be given. Then

(A)

tt

c

ts

(l)dlds

c

2M2

and

(B)

d

dt

ttc

ts

(l)dlds=c(t) t

tc(r)dr

hold for all t

R.

3 Statement of main result and remarks

For simplicity, we assume that our system (1) has the form

x=f(x, t,p(t/)) (4)

wherep: R Rd (for some integerd) is bounded and its componentsp1, . . . ,pdhave bounded first derivatives. We set

p:=sup{|p(r)|: r R} and R(p):= {p(t): t R}.

Our next assumption is a variant of those of [13, Theorem 2] (but see Sect.5for results under weaker assumptions).


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Assumption 5 There exist1, 2 K, positive constantsca,cb, andT, a con-tinuous functionq : Rd R, and aC1 functionV: Rn [0, ) Rd [0, )such that

A1 1(|x|)V(x, t, )2(|x|),A2 Vt(x, t, ) + Vx(x, t, )f(x, t, ) q( )V(x, t, ),A3 |V(x, t, )| caV(x, t, ), andA4

ttTq(p(s))dscb

hold for allx Rn,t0, and R(p).Note thatA2 is weaker than the standard exponential stability property of

the frozen dynamics since we do not require1or2to be quadratic functions

and moreoverq( )can take nonpositive values for some choices of the vectorparameter . However, A4 requires that q be positive on average along thevectorp(s).

Theorem 6 If(4)satisfies Assumption5, then for each constant > 2Tca p/cb,the dynamics(4)are UGAS and

V(t,x):

=e

T

t

t T

t

sq(p(l))dlds

V(x, t,p(t/)) (5)

is a Lyapunov function for(4).

Remark 7Compared with the known results [6, 13,14], the novelty of Theorem6is that (a) we allow fully nonlinear systems including cases where the functionqcan take both positive and negative values (which corresponds to the allow-ance in [14] of eigenvalues that wander into the right half plane while remainingin the strict left half plane on average) and (b) we provide an explicit Lyapunovfunction (5) for the original slowly time-varying dynamics. In general, the con-

clusion of Theorem6may or may not hold for small values of . We illustratethis in Sect.6.

Remark 8 A completely different approach to slowly time-varying systemsx=f(x, t, t/) (for large constants >0) is to transform the system into a rapidlytime-varying system and to then try to construct a Lyapunov function for theresulting rapidly time-varying system directly. The transformation is done bysimply settings=t/which gives rise to the new rapidly time-varying system

x(s)

=g(x(s),s, s):

=f(x(s),s,s) (6)

in terms of the new rescaled time variable s. However, it might be difficultto apply the Lyapunov function construction methods of[11] or other knownmethods to build an explicit Lyapunov function for(6). This is because theseearlier results are for fast time-varying dynamics having a different form from


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(6) and moreover they require Lyapunov functions for so-called limiting dynam-ics; cf. [12, Property 2], and see [11, Sect. 3.1] for a generalization in the samevein. This motivates our direct construction of Lyapunov functions for slowlytime-varying dynamics, which may be viewed as a complementary approach to

the time rescaling method since we do not require limiting dynamics.

4 Proof of Theorem6

ByA2A3and our choice ofp, the time derivative of

V(x, t):

=V(x, t,p(t/)) (7)

along the trajectories of (4) satisfies:

V= Vt(x, t,p(t/)) + Vx(x, t,p(t/))f(x, t,p(t/)) + V(x, t,p(t/))

p(t/)

q(p(t/))V(t,x) + V(x, t,p(t/))p(t/)

q(p(t/)) + ca p

V(x, t).To simplify the notation, let us define

E(t, ) :=eT

t

t T

t

sq(p(l))dl

ds

. (8)

Sincepis bounded andqis continuous,

(t):=q(p(t)) (9)

is bounded in norm by some constantM>0. Therefore, along the trajectoriesof(4), Lemma4(B) with the choices (9) andc=Tgives

V= E(t, )

V+

q(p(t/))

1

T

t

tT

q(p(l))dl

V

.


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Substituting the formula forV, it follows fromA4that

V

E(t, ) ca p 1Tt

tT

q(p(l))dl V

E(t, )

ca p

cbT

V(x, t). (10)

Applying Lemma4(A) with the choices (9) andc=Tgives

eTM/2 E(t, )eTM/2

everywhere. Hence, for >2Tca p/cb, (10) gives

V (x, t) cb

2TeTM/2V(x, t) 3(|x|), (11)

where3(s)= cb2TeTM/21(s)is positive definite; and

1(|x|)V (x, t) 2(|x|) (12)

everywhere, where

1(s):=eTM/21(s), 2(s):=eTM/22(s)

are of class K. Since (11)(12) imply thatV is a Lyapunov function for (4),

Theorem6follows from Lemma3.

5 More general families of Lyapunov functions

We next show how to relax requirementsA2A3from Theorem6.We continueto use the notation we introduced in Sect.3. We assume the following in therest of this section:

Assumption 9 There exist1,2 K, a positive definiteC1 function, posi-tive constants T, ca,and cb, a continuous functionq: Rd R,anda C1 functionV : Rn [0, ) Rd [0, )such that

limr+

r1

1

(l)dl= + (13)

and


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A1 1(|x|)V(x, t, ) 2(|x|),A2 Vt(x, t, ) +Vx(x, t, )f(x, t, ) q()(V(x, t, )),A3 |V(x, t, )| ca(V(x, t, )), andA4 ttTq(p(s))ds

cb

hold for allx Rn,t0, and R(p).Notice that Assumption5is the special case of Assumption9in which (l)

l. We prove the following:

Theorem 10 If(4) satisfies Assumption 9, then there exists k K for whichthe requirements of Assumption 5 are satisfied with V := k(V). Therefore, foreach sufficiently large value of the constant > 0, the dynamics(4) is UGASand admits a Lyapunov function of the form(5).

It suffices to prove the first statement of Theorem10since the second state-ment is immediate from Theorem 6. To this end, we use the following importantobservation:

Lemma 11 IfC1 is positive definite, then

limr0+

r1

1

(l)dl= . (14)

Proof SinceC1 and(0)=0, we can findc3 >0 such that(r)c3rforallr [0, 1]. Hence, for eachr(0, 1], we get

1r

1

(l)dl

1r

1

c3ldl= 1

c3ln(r). (15)

It follows that, for allr(0, 1],

r1

1

(l)dl 1

c3ln(r). (16)

Sincelimr0+ln(r)= , the lemma follows. Given a constant >0 which we specify later, it follows from (13) and (14)

that

k(r)= e

r

1

1

(l)dl

, r> 00, r=0

(17)

is continuous and unbounded. In particular, k(r) 0 as r 0+. Set B=sup{(s): 0s1}, which is positive since is positive definite.


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Lemma 12 The function(17)is C1 when= 2B.Proof It suffices to prove that

k(r)0 as 0< r0+, (18)sincek(0)= 0, because thenk(r)/r 0= k(0)asr 0+. To this end, firstnote that

k(r)= (r)

e

1r

1(l)

dl, r> 0 (19)

and that for allr(0, 1], we have

1B

ln((1)) ln((r)) =

1r

(l)B(l)

dl1

r

1(l)

dl

by our choice ofB. Since is positive definite andis positive, this implies

(r)e

B [ln((1))ln((r))]

(r)e

1r

1(l)

dl= k(r)

for allr(0, 1], i.e.,((1))

B (r)

B1 k(r) 0 r(0, 1] (20)

so (18) follows from our choice of. The fact that kKC1 is now immediate from Assumption 9 and Lemmas

11and12. Let us now verify thatksatisfies the requirements of Theorem10.From the definition ofk, we deduce that

k(V)(V) = 2B(V)

e2B 1

V

1(l)

dl

(V) = 2Bk(V)

whenV= 0. Therefore, by assumption A2, the time derivative ofV= k(V)along the trajectories of (4) satisfies

Vt(x, t, ) + Vx(x, t, )f(x, t, )=k(V(x, t, ))[Vt(x, t, ) +Vx(x, t, )f(x, t, )]

k(

V(x, t, ))

q()(

V(x, t, ))

= 2Bq( )V(x, t, )

everywhere, and conditionA3from Assumption9implies

|V| = k(V)|V| cak(V)(V) = 2BcaV


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everywhere. Therefore, Assumption5holds using

i(s):=k i(s) fori=1, 2, V:=k(V), q( ):=2Bq( ),ca :

=2B

ca, cb :

=2B

cb. (21)

The result now follows from Theorem6.

6 Examples

We illustrate our constructions using four examples. In the first two examples,the functions V from Assumption 5 do not depend on the frozen parame-ter , so we can conclude that (4) is UGAS for all values of the constant

>0. We then turn to a slowly time-varying friction dynamics and an examplefrom identification where V depends on , and where we can consequentlyonly conclude the UGAS property of(4) when > 0 is sufficiently large. SetV(x, t, ):=Vt(x, t, ) + Vx(x, t, )f(x, t, )everywhere.

6.1 Stability for all >0: a scalar example

Consider the one-dimensional system

x= x1 +x2

1 90 cos2

t

(22)

and the uniformly proper and positive definite function

V(x, t, )V(x):=e

1+x2 e. (23)

Let us verify Assumption5for this choice ofVand the frozen dynamics

x=f(x, t, ):= x1 +x

2

1 90 , 0 1. (24)

This gives

V(x, t, )=e

1+x2 x2

1 +x21 90

=e

1+x2 x

2

1 +x2

90e

1+x2 x

2

1 +x2

. (25)

Simple calculus calculations everywhere give

2e

2

e 1 V(x) x2

1 +x2 e

1+x2 12

V(x) (26)


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so (25) everywhere gives

V(x, t, ) 2e

2

e

1 45V(x). (27)

Moreover, for eacht0, we gett

t

45 cos2(s) 2e

2

e 1

ds =

45

2 2e

2

e 1

> 0

which shows that Assumption5is satisfied. We conclude from Theorem6thatfor large enough constants >0, (22) is UGAS and has the Lyapunov function

e

t

t

t

s

45 cos2(l) 2e

2

e

1

dl

ds

V(x)

=e454

sin( 2t

)+ 4 e

2

45(e1)

[e

1+x2 e] (28)

whereVis in (23). In fact, since in this caseVdoes not depend on, it followsfrom our proof of Theorem6that for any constant > 0, the system (22) is

UGAS and admits the Lyapunov function (28).Remark 13 The dynamics (22) illustrates the fact that our approach applies tosystems which are not globally exponentially stable. Indeed, it is clear that (22)is not globally exponentially stable since its vector field is bounded in norm bythe constant 91.

6.2 Stability for all >0: a pendulum example

Our constructions also apply to the slowly time-varying pendulum dynamics[13]x1=x2x2= x1 [1 + b2(t/)m(x, t)]x2

(29)

assuming

(P1) m: R2 R [0, 1]is Lipschitz continuous; and(P2) b2 : R(, 0]is globally bounded, and there are positive constants

Tandcbsuch that 5 + T

ttTb2(l) dlcbfor allt R.

The dynamics (29) was shown to be UGAS for certain choices of the functionb2 in [12]; see [6] for related results, and [14] for results that are restricted tothe linear case. However, these earlier results do not lead to explicit Lyapunovfunctions for (29). In order to build Lyapunov functions for (29) for large con-stants > 0, we use the following observation for the corresponding frozendynamicsf(x, t, ):=(x2, x1 [1 + m(x, t)]x2):


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Lemma 14 The function V(x):=x21 +x22 +x1x2satisfies V(x)f(x, t, ) [1+5]V(x)for all x Rn, t R, and 0.ProofBy grouping terms, one easily shows that

V(x)f(x, t, )= V(x) 2m(x, t)x22 m(x, t)x1x2 (30)

everywhere. Since

V x21+x22 |x1x2| 1

2x21+

1

2x22 |x1x2|

everywhere, we get

2m(x, t)x22 4m(x, t)V(x), m(x, t)x1x2 m(x, t)V(x) (31)

everywhere. The lemma follows by substituting (31) into (30) and recalling that0m(x, t)1 everywhere.

The following is an immediate consequence of Lemma14,the proof of The-orem6, and the fact thatV(x):=x21+x22+x1x2only depends onx:Theorem 15 Let(29)satisfy(P1)(P2). Then(29)has the Lyapunov function

V (t,x):=e5T

t

t T

t

sb2(l)dlds

(x21+x22+x1x2) (32)

for each choice of the constant > 0. Hence, (29) is UGAS for all constants >0.

6.3 Friction example revisited

We next illustrate Theorem6using the one degree-of-freedom mass-spring sys-tem [4, 11]. The mass-spring system arises in the control of mechanical systemsin the presence of friction. However, in contrast to [11] where the dynamicsare assumed to be rapidly time varying, here we consider the case where thedynamics areslowlytime varying. While slowly time varying dynamics can betransformed into rapidly time-varying dynamics by rescaling time, doing so forthe slowly time-varying mass spring system produces a new dynamic that doesnot lend itself to the known methods; see Remark8for details. For this reason,we directly apply the slowly time-varying theory we developed in the precedingsections.

Let us recall the model[11]. The dynamics are given by

x1= x2x2= 1(t/)x2 k(t)x1

2(t/) + 3(t/)e1(x2)

sat(x2)

(33)


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wherex1 andx2 are the mass position and velocity, respectively;i :[0, )(0, 1], i= 1, 2, 3, denote positive time-varying viscous, Coulomb, and staticfriction-related coefficients, respectively;1 is a positive constant correspond-ing to the Stribeck effect; (

) is a positive definite function also related to

the Stribeck effect; k denotes a positive time-varying spring stiffness-relatedcoefficient; and sat()denotes any continuous function having these properties:

(a) sat(0)=0, (b) sat( )0 R,(c) lim

+sat( )= +1, (d) lim

sat( )= 1 (34)

Following [11], we model the saturation differentiably as

sat(x2)=tanh(2x2), (35)

where2 is a large positive constant, so|sat(x2)| 2|x2|for allx2 R. How-ever, unlike [11], we assume that the friction coefficients vary in time slowerthan the spring stiffness coefficient so we restrict to cases where >1. We aregoing to establish the stability of (33) and construct corresponding LyapunovfunctionsV when the constant >0 is sufficiently large.

Our precise mathematical assumptions on (33) are: k and the is are C1 func-

tions with bounded derivatives;has a globally bounded derivative; and thereexist constantscb, T>0 such that

ttT

1(r)dr cb t0. (36)

We also assume this additional condition whose physical interpretation is that

the spring stiffness is nonincreasing:

ko,k>0 s.t. kok(t)k and k(t)0 t0.

The frozen dynamicsx=f(x, t, )for (33)are

x1= x2x2= 1x2 k(t)x1

2+ 3e1(x2)

sat(x2)

(37)

where = (1, 2, 3) [0, 1]3 is now a vectorof parameters. We apply ourconstruction from Theorem6withp(t)=(1(t), 2(t), 3(t))and the function

V(x, t, )=A(k(t)x21+x22) + 1x1x2 whereA=1 +k0

2+ (1 + 22)

2

k0. (38)


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We first verify the conditions of Assumption5.SinceA max{1, 1/ko}and11, we have

1

2(x2

1+x2

2)

V(x, t, )

A2

k(

|x1

| + |x2

|)2

2A2

k|x

|2 (39)

everywhere. Let us now compute V(x, t, ) for all values [0, 1]3. Sincek(t)0 everywhere, this gives

V(x, t, )Vx(x, t, )f(x, t, )= [2Ak(t)x1+ 1x2]x2 [2Ax2+ 1x1]{1x2

+

2+ 3e1(x2)

sat(x2) + k(t)x1}.

Therefore, by grouping and canceling terms, we also have

V(x, t, ) 1k0x21 (2A1 1)x22+ 1(1 + 22)|x1x2|

1ko

2|x|2

1

ko

2 x21+ (A 1/2)1x22 1(1 + 22)|x1x2|

= 1ko

2|x|2 1

ko

2

|x1|

1 + 22ko

|x2|2

+1(1 + 22)2

2ko +1

2A1x22 1ko

4A2kV(x, t, )

where the first inequality follows from (34)(b), the inequality |sat(x2)| 2|x2|,and the fact thati [0, 1]for eachi; the second inequality uses the fact thatA 1/2k0/2; and the last inequality is from the choice ofAand the bounds(39). Hence, Assumption5of Theorem6readily follows from (36) with thechoices

q( )= 1k04A2k

, ca=1.

We conclude as follows:

Corollary 16 Under the preceding assumptions, there exists a constant0 > 0such that for all constants > 0, the system (33) is UGAS and admits theLyapunov function

V (t,x):=V(x, t,p(t/)) ebT

t

t T

t

s1(l)dlds

(40)

where V is the function defined in(38),b=k0/(4A2k), and p(t)=(1(t), 2(t),3(t)).


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6.4 Identification dynamics revisited

Our Lyapunov function constructions also apply to the slowly time-varyingdynamics

x=h(t/)m(t)m(t)x, x Rn (41)assuming that there are positive constantsT,c,, andsuch that(I1) h : R [, 0] is continuous with a bounded first derivative andt

tTh(r)dr for allt R.(I2) m: R Rn is continuous and satisfies |m(t)|1and It+ct m(r)m(r)

dr Ifor allt R.whereIis the identity matrix and for matricesA, B Rnn we useB A0to mean thatB Ais positive semi-definite. See also Remark23for the gener-alization of our result to control affine systems

x = h(t/)m(t)m(t)x+ g(x, t, t/)u

for suitable matrix valued functionsg. The particular casex= m(t)m(t)xof(41) has been extensively studied in the context of identification theory[11, 12].In [11], we showed how to construct explicit Lyapunov functions for the rapidlytime-varying systemx=f(t)m(t)m(t)x for appropriate nonpositive functionsfand large positive constants. However, these earlier results do not lead toexplicit Lyapunov functions for the slowly time-varying dynamics (41) for largeconstants > 0; see Remark8. Instead, we construct Lyapunov functions for(41) using the following analogue of[11, Lemma 6]:

Lemma 17 Assume that there are positive constants T,c, , and such that(I1)(I2)are satisfied and set

P(t, )=I t

tc t

s m(l)mT(l) dlds, where=c

2+

2c4

2+c2. (42)

Then for each [, 0]the function

V(x, t, ) = xP(t, )x

satisfies the requirements of Assumption5for the frozen dynamics f(x, t, )=m(t)m(t)x and p(s)=h(s).Proof We first apply Lemma 4(B) to (t)

= xm(t)m(t)x for each x and

[, 0]to get

V

t (x, t, )= cxm(t)m(t)x + x

ttc

m(l)m(l)dl

x


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and

V

x

(x, t, )f(x, t, )=

2x I t

tc

t

s

m(l)mT(l) dlds m(t)m(t)x

so the derivativeV= Vt

+ Vtfalong the trajectories offsatisfies

V [(2c)|m(t)x|2 + |x|2] + 2|x||m(t)x|c2= {(2c)|m(t)x|2 + |x|2 + |x||m(t)x|c2} (43)

where the inequality follows from Lemma 4(A), (I2), and the facts that|m(t)

| 1 and 0. Set

= 2c2

.

Then the triangle inequality gives

|m(t)x||x| |x|2 + 14

|m(t)x|2

so since 0, we get

(2c)|m(t)x|2 + |x|2 + |x||m(t)x|c2

(2c)|m(t)x|2 + |x|2 + c2|x|2 + c2

4|m(t)x|2

=

2c + c2

4

|m(t)x|2 + (+ c2)|x|2

|x|22

by our choices of and . This and (43)give V 2|x|2 everywhere; andLemma4(A), the fact that|m(t)| 1, and our choice of give

|x|2 V(x, t, ) (+c2)|x|2

V

(x, t, ) c2

2|x

|2

V(x, t, ).

Hence,

V(x, t, ) q( )V(x, t, ), where q( ):= 2(+c2)


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everywhere. Therefore, we can satisfy the requirements of Assumption5usingp(s)=h(s)and

1(s)=s2

, 2(s)=(+c2

)s

2

, ca=1, cb= 2

2(+c2) . (44)

This proves the lemma.

The following is an immediate consequence of the preceding lemma andTheorem6:

Theorem 18 Let(41)admit positive constants T,c,, andsuch that(I1)(I2)are satisfied and choose cb as in(44). Then for any constant2Tsup{|h(r)|:r

R

}/c

b, the function

V (t,x):=e

2T(+c2)

t

t T

t

sh(l)dlds

V(x, t, h(t/)) (45)

with V(x, t, )=xP(t, )x as defined in Lemma17 is a Lyapunov function for(41). Hence,(41)is UGAS for all constants 2Tsup{|h(r)|: r R}/cb.

7 Input-to-state stability

We next extend our results to control affine slowly time-varying systems

x=f(x, t,p(t/)) +g(x, t,p(t/))u (46)

evolving on Rn with control valuesu Rm, wheref : Rn [0, ) Rd Rnand g : Rn [0, ) Rd Rnm are locally Lipschitz functions that admit4 Ksuch that

|f(x, t,p(t/))| + |g(x, t,p(t/))| 4(|x|)

everywhere, and wherep : R Rd for somedis bounded with a bounded firstderivative. The control functions (i.e. inputs) for (46) comprise the set Uof allmeasurable essentially bounded functionsu :[0, ) Rm with the essentialsupremum norm| |. We assume throughout this section that Assumption5holds for some functionV C1 and that1 K andca from Assumption5are also such that

A5 |Vx(x, t,p(t/))| ca1(|x|)A6 |g(x, t,p(t/))| ca

1 + 41(|x|)

hold for allt0, >0, andxRn. Notice thatA5reduces to a linear growthcondition when 1(x)= |x|2 and so automatically holds in the classical casewhereVhas the formxP(t)xfor a suitable bounded positive definite matrix.


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We show that when Assumption5and A5A6 hold, and when the constant > 0 is sufficiently large, the control system(46) satisfies the input-to-statestable (ISS) property and admits the ISS Lyapunov function (5). We first recallthe relevant ISS definitions from [5,8,15,16].

For a general locally Lipschitz control affine system

x=h(x, t) +J(x, t)u (47)

where h andJare uniformly state bounded (as defined in Sect.2), and for givenvaluest0 0,x0 Rn, andu U, we lett (t; t0,x0, u)denote the uniquemaximal solution of the initial value problem

x(t)=h(x(t), t) +J(x(t), t)u(t) a.e. t, x(t0)=x0.

We always assume all trajectories(; t0,x0, u)so defined are defined on all of[t0, ). Later we specialize to the controlled system (46) for fixed constants >0.

Definition 19 We say that (47) is ISS provided there exist KL and Ksuch that

| (t; t0,x0, u)| (|x0|, t t0) + (|u|) (ISS)

holds for alltt0,t00, x0 Rn, andu U.Definition 20 A smooth function W : Rn [0, ) [0, ) is called an ISSLyapunov function for (47) provided there exist functions1, 2, Kand apositive definite function3such that

(1) 1(|x|)W(x, t)2(|x|) and(2) |u| (|x|)impliesWt(x, t) + Wx(x, t)[h(x, t) +J(x, t)u] 3(|x|).hold for allx Rn,t0, andu Rm.

The following lemma comes from [5]:

Lemma 21 If(47)has an ISS Lyapunov function, then it is ISS.

We prove the following analogue of[11, Theorem 4]:

Theorem 22 Assume that(4)satisfies Assumption 5. Assume A5A6everywherehold where ca, V, and1are chosen as in Assumption5.Then for each constant >4Tca p/cb, the dynamics(46)are ISS and

V (t,x):=eT

t

t T

t

sq(p(l))dlds

V(x, t,p(t/)) (48)

is an ISS Lyapunov function for(46).


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Proof We indicate the changes needed in the proof of Theorem6.Consider thefunction Kdefined by

(s):= cb

1(s)

2Tc2a

1 + 41(s) ,where 1 and ca are as in Assumption5. This function is of class K sinces 1/41 (s) ands s2/(1+s) are both K. Our assumptions imply that if|u| (|x|), then

|Vx(x, t,p(t/))g(x, t,p(t/)u| cb1(|x|)

2T

cb

2TV(x, t,p(t/))

everywhere. DefineVas in (7) andE(t, )as in (8). Then, along any trajectoryx= (t)of (46) with inputsusatisfying|u| (|(t)|)everywhere, we get

V= q(p(t/))V(t,x) +

ca p

+ cb

2T

V(x, t)

everywhere and therefore (by reasoning exactly as in Sect.4) also

V E(t, )

ca p

+ cb

2T 1

T

t/t/T

q(p(l))dl

V(x, t)

E(t, )

ca p

cb2T

V(x, t) cbE(t, )

4TV(x, t)

when >4Tca p/c

b, where the second inequality follows fromA

4, and the last

inequality follows from our choice of. We then argue exactly as before to show

thatV is an ISS Lyapunov function for (46) for all constants > 4Tca p/cb.

The theorem now follows from Lemma21. Remark 23 Theorem22readily applies to our friction example from Sect.6.3,showing that (40) is actually an ISS Lyapunov function for the slowly time-varying controlled friction dynamic

x1

=x2

x2= 1(t/)x2 k(t)x1+g(x, t, t/)u 2(t/) + 3(t/)e1(x2) sat(x2)(49)

for anyg satisfying our assumptionA6 for someca >0 and with1(s)=s2/2,provided that the constants ca > 0 and > 0 are sufficiently large. Similarextensions can be made for the other examples we considered above.


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8 Conclusion

We provided general conditions under which slowly time-varying systems areuniformly globally asymptotically stable and ISS with respect to general per-

turbations, thus extending[13, 14] to situations where the corresponding frozendynamics are not necessarily exponentially stable. Moreover, we provided newmethods for constructing explicit closed form strict ISS Lyapunov functions forslowly time-varying control systems in terms of a suitable family of generalizedLyapunov like functions for the frozen dynamics. This is significant becauseLyapunov functions play essential roles in robustness analysis and controllerdesign.

We conjecture that our work can be extended to systems that are subjected toactuator or measurement errors, or which are components of larger controlled

hybrid dynamical systems. It would also be of interest to extend our work toslowly time-varying systems with outputs and to construct corresponding input-to-output stable (IOS) Lyapunov functions; see [17, 18] for further backgroundon systems with outputs and [7] for some first results on constructing IOSLyapunov functions for nonautonomous systems in terms of given nonstrictLyapunov functions. We leave these extensions for future papers.

Acknowledgments This research was supported by NSF Grant 0424011. This work was done whilethe first author visited Louisiana State University (LSU). He thanks LSU for the kind hospitalityhe enjoyed during this period. Both authors thank Marcio de Queiroz and Patrick De Leenheer

for illuminating discussions.

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